On quantum gravity problem within geometric approach D.G. Pak - - PowerPoint PPT Presentation

On quantum gravity problem within geometric approach

D.G. Pak JINR, LTP

• Int. Workshop “Bogolyubov Readings
• Sept. 2010

Plan of talk

* Preliminaries: geometric approach to quantum gravity problem

• I. Vacuum tunneling in Einstein gravity.
• II. Quantum gravity models with torsion

Main principles, ideas Yang-Mills type gravity model Induced effective Einstein gravity

• III. Minimal model of Lorentz gauge gravity with torsion

Basic idea, the model Dynamical content in Lagrange formalism Covariant quantization * Conclusions

Preliminaries

Quantum Gravity – Great Puzzle – numerous approaches:

Quantum Einstein gravity may exist non-perturbatively Classical geometric generalizations including torsion, Weyl fields, non-metriicty Canonical gravity approach Loop gravity Spin foam models Super-gravity, -strings, M-theory Extra-dimensions, brane worlds Non-commutative geometry Analogous models in condensed matter physics ……………………… Gravitation is manifestation of geometry and it does not exist as a fundamental force at all ----- an extreme viewpoint

• I. Vacuum tunneling in Einstein gravity

Definitions of vacuum: flat space-time (i) an absolute vacuum

Fubini-Study instanton CP2 describes vacuum-vacuum transition

0:

ijkl ij ij

R g η Λ = = → =

ij

1 0: R 2

ij ij ij

Rg g g Λ ≠ − + Λ = → =

0 when

mn

g t → → ±∞

(ii) Any static vacuum solution

↓

1. 2. Asymptotically flat metric with space topology S3 or RP3

0: Λ ฀

this type of vacuum metrics corresponds to asymptotically locally Euclidean instantons (ALE)

3.

3, 1 χ τ = =

On Fubini-Study instanton on CP2

2 2 2 2 2 2

4 ( ), , 1 1 0 , 1 1 3 3, 1, 2

m n m n mn mn n m mn mn

x x x x a g a a x C x C a δ ρ ρ χ τ + = − + + = ⎛ ⎞ ⎜ ⎟ − ⎜ ⎟ = ⎜ ⎟ ⎜ ⎟ − ⎝ ⎠ = = Λ = % % %

5 * 5

1 ( ) 4

a a

j ge RR

µ µ µ µ

ψγ γ ψ ∂ = ∂ =

Axial ABJ anomaly The anti-instanton is given by the same metric but with opposite orientation of the space-time, i.e. with inversed vielbein /Egichi&Freund, PRL’1976/ An analog to BPST instanton

Topological structure of the vacuum: the main assumption:

* The vielbein (tetrad) eam is a fundamental variable of quantum gravity, not a metric tensor. The non-trivial topology is provided by In Euclidean gravity the vacuum is classified by two integers (m, n) since SO(4)~SU(2)xSU(2)

• S. Hawking (1978):

There is no vacuum tunneling due to infinite /Cargese lectures procs./ barrier between vacuums.

3 3

( (1,3)) ( (3)) SO SO π π = = ฀

3 3

( (4)) ( (2) (2)) SO SU SU π π = × = ⊗ ฀ ฀

In search of instantons: a simple ansatz

• A pure gauge vielbein and pure

gauge spin connection are:

( ) ( ) ( )

am ab bm am mcd ce m ed

e x L x L x L L δ ϕ = = = ∂ %

In temporal gauge the topological vacuums are classified by Chern-Simons number

3 2

1 ( ) 16

mcd CS CS

N Tr d xω ϕ π =

∫

A simple hedgehog ansatz with an arbitrary trial function “g” produces a class of conformally flat metrics

ˆ 2 2 2

( ) ˆ / , tan / ,

i i

n x n ma ma ma

x e g e x x r r t r t

ωη

ρ δ ρ ω ρ = = Θ = = = +

n ma

Θ

is a generalization of ‘t Hooft matrices

i ma

η

For the vanishing Ricci scalar

• ne obtains a simple regular solution -
• Hawking wormhole :

3 ' 2 ( '' ) g R g g ρ = + =

2 2

( ) 1 g λ ρ ρ = +

2 2 2 2 2 2

4 ( 4 ) ( )

mn mn m n ijkl

R x x C λ δ ρ ρ ρ λ τ = − + = → =

Ricci and Weyl tensors are: The solution can be interpreted as instanton-antiinstanton pair in conformal gravity No other appropriate instanton solutions (asymptotically locally flat and with non-zero Hirzebruch signature) were found

In search of instantons: 1-1 correspondence between topological vacuums in Yang-Mills theory and gravity

• The spin connection can be decomposed into rotation and boost parts

mcd

ϕ

3 3 3

(2) (3) (1,3) ( (1,3)) ( (3)) ( (2)) SU SO SO SO SO SU π π π ≈ ⊂ = =

)

(

,

mcd m m

B ϕ = Ω

Since

• ne can construct vacuum spin connection

in terms of SU(2) gauge potential

)

(

( ) ,0

vac mcd m

e ϕ = Ω

By this way it is not easy to retrieve the vacuum vielbein from the spin connection.

Explicit construction of non-trivial topological vacuums in SU(2) Yang-Mills theory /Baal&Wipf-2001, Cho-2006/ Introduce orthonormal basis of SU(2) triplets

ˆ ˆ : 0, 1,2,3 (2)

i m i

n D n SU

α

α = = ∈

One has the integrability condition

ˆ ˆ [ , ]

m n i mn i

D D n F n = × = r

Solution to these conditions gives a pure gauge vacuum potential /Cho-2006/

ˆ , 1 ˆ ˆ ( ) 2

k m m k k k m ij i j

C n C n n ε Ω = − = − ⋅ r

Parameterizing the triplet by angles of S3~SU(2)

• ne obtains explicitly:

ˆi n

1 2 3

sin sin cos , cos sin sin , cos

m m m m m m m m m

C C C γ α α γ β γ α α γ β α β γ = ∂ − ∂ = ∂ + ∂ = ∂ + ∂

In 4d spherical coordinate system the radial coordinate hypersurfaces are given by S3. So one can define the basis triple of left invariant 1-forms on S3

1 , 2 2

i m i m i ijk j k

dx C d σ σ ε σ σ = =

Maurer-Cartan eqn.

Finally, the basis of pure gauge vielbein 1-forms in polar c.s. is defined as follows:

( , )

a i

e dρ ρσ =

1 ( , , ) 2

i m i m

dx C σ α β γ =

where the angle functions

( , , ), ( , , ), ( , , ) α θ φ ψ β θ φ ψ γ θ φ ψ

( , , , ) ρ θ φ ψ

define the homotopy classes

3(

(2)) SU π

To find instanton solutions one can apply a simple ansatz:

( ( ) , ( ) )

a i i

e g d g ρ ρ ρ ρσ =

We will consider an ansatz corresponding to topological class with winding number

1, i.e., we put , , τ α θ β φ γ ψ = = = =

The ansatz with two functions g0 , g3 (g1=g2=1) applied to Einstein eqn. produces the well-known Eguchi-Hanson instanton

2 2 2 4 3

1/ 1 / g g a ρ = = −

Explicit proof of vacuum tunneling

Passing to temporal gauge in Cartesian coords. gives a system

• f eqs. for gauge parameters

1 1 2

ˆ ˆ exp[ ( , ) ( , )], 1

t t t i i

A UAU U U U i r t f r t f ω τ

− −

→ + ∂ = = = r r r

1,2,3

( ) ( )

i i m i i m

e g dx A x ρ ρσ

=

= =

The space-like vielbein of E-H instanton defines SU(2) gauge potential Ai

m:

ˆ , f ω In asymptotic region t

±∞ ฀

• ne has the solution

3 1 2

ˆ 1, ( ) rg f dt c r ω ρ +

∫

฀ ฀

where c1 is determined by initial condition

( ) t ω = −∞ =

This implies transition from the trivial vacuum defined by

ˆ( ) (0,0,1) n t = −∞ =

to non-trivial vacuum with NCS=1 at

t = +∞

defined by

sin ( )cos ( ) ˆ ˆ sin ( )sin ( ) cos ( )

t t t

r r n U n r r r α β α β α

=+∞ =+∞ =−∞

⎛ ⎞ ⎜ ⎟ = − = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

where the functions ,

α β are defined by

ˆ ˆ ( , ) exp[ ( ) ( )], 2 ˆ ( ) (sin ( ),cos ( ),0).

i i t i

i U f r r r r r ω α τ β β β β

=+∞

= = r 1|

CS CS vac

N N < = = >

inst

S

e− ฀

Vacuum tunneling

2, 1 χ τ = =

via Eguchi-Hanson instanton via Fubini-Studi instanton

( )

mn

g t = ±∞ =

CP2

\

3, 1 χ τ = = Λ ≠

Λ ฀

What is strange in this vacuum tunneling?

The more principal question is:

Whether vielbein really represents a variable of quantum gravity?

* Is the vielbein like a kinematic variable locally introduced on water surface? If this is so, what describes the microscopic structure of the space? * testing the quantum nature via gravitational Aharonov-Bohm effect: calculation of holonomy operator and experimental verification. * 1979: Hawking’s claim was rather limited to asymptotic euclidean instanton. * 1979: Why others did not claim the vacuum tunneling? * 2008: Vacuum tunneling revisited /Y.M. Cho, Prog.Th.Phys.Suppl.,2008/ * ~1920s: Schwarzschild prefers RP3 as more simple than S3 * Indications to RP3 topology of our space:

• -non-zero index I3/2 of Dirac operator for E-H instanton;
• - existence of the electrons.

• II. Quantum gravity models with torsion

Why torsion (contortion, Lorentz connection)?

* Equivalence principle, local Lorentz symmetry, gauge principle.

If the vielbein is classic then the quantum fluctuation

• f spin connection will create general Lorentz connection

fluctn mcd

( )

mcd e

A ϕ ⎯⎯⎯ →

* Einstein gravity as effective theory induced by quantum dynamics.

Contortion (torsion) may provide the microscopic structure of the space

* Existence of spin particles should imply torsion.

A problem of non-existency of solution for the electron in Einstein gravity

* Ideas from QCD: confinement, quantum condensate

Torsion might be unobservable as a classic object like gluon in QCD--Quantum chromodynamics, there is no classical chromodynamics.

* Contortion should possess properties of connection.

Contortion as a part of Lorentz connection, not a tensor.

Yang-Mills type Lorentz gauge gravity

Utiyama gauge approach to gravity Riemann-Cartan curvature : The Lagrangian is : This Lagrangian admits an additional local symmetry:

cd cd cd

K e A

µ µ µ

ϕ + = ) (

d e b e c a d c b a mcd abcd mcd abcd abcd abcd

K K K D K e R K e R R R

] | [ ] [

ˆ ) ; ( ~ ) ; ( ~ ˆ + = + =

abcd abcdR

eR L 4 1 − = ], , [ ˆ ; Λ + Λ = = =

µ µ µ µ µ

δ δϕ δ K D K e

cd a

One-loop effective action in constant curvature space-time background

• Splitting the contortion into

classical and quantum parts and assuming the vacuum averaged value for Riemann-Cartan curvature to be positively constant

• ne can calculate a one-loop effective

potential which has a non-trivial minimum leading to torsion condensate: /Class. Quant. Grav.,2008/

cd class cd cd

Q K K

µ µ µ

+ =

) ( ~

2 bc ad bd ac abcd

M R η η η η − >= < ) ~ (ln 48 ~ 11 3 ~

2 2 2 2 2

c R g R g R Veff − + = µ π

Induced Einstein gravity as an effective theory

Expanding the initial Lagrangian around vacuum one obtains the effective Lagrangian with Einstein Hilbert term and cosmological constant The value of M is related with the minimum of the effective potential and vacuum energy density.

Weak coupling phase: Strong coupling phase:

this value is close to coupling constant in SO(10) SS GUT at unification scale 1017 Gev

4 2 2

2 3 ˆ 2 1 2 ^ ˆ 4 1 ) ~ ˆ ( 4 1 4 1 M M R R R R e R eR L

abcd abcd abcd abcd

− − − = = > < + − = − =

012 . 4 10 6 . 3 10 2

2 2 24 2 4 47

≈ = × = × ≈

− −

π α ρ g Gev M Gev

vac

5 . 1 , 1 ≈ ≈ Λ α

Problems, drawbacks of the Yang-Mills type gravity model * The Hamiltonian is not positively defined; * Two independent variables, vielbein and contortion on equal

• footing. Formally the vielbein should be quantized also. But

the fundamental quantum variable should be rather only one. * The content of spin states of contortion is too big to compare with vielbein: two spin 2 states, four spin 1 states implies 24 degrees of freedom;

(i) Dynamical symmetry breaking, Meissner effect, confinement, gluon condensate (ii) A vacuum pure gauge potential is described by SU(2) gauge potential * Meissner effect with forming torsion condensate (Regge&Hanson’80s), torsion as a genuine connection can be confined * Flat vielbein is represented by the same pure gauge SU(2) potential (in spherical coordinates)

) ˆ ˆ ( 2 1 , ˆ

j i ijk k i k k i

n n C n C A

µ µ µ µ

ε ∂ ⋅ = =

i i i a

C dx where d e

µ µ

σ ρσ ρ 2 1 ), , ( = =

Possible interrelation of QCD and Gravity

QCD Gravity

µ µ µ

X A A r r + = ˆ

cd cd cd

K e A

µ µ µ

ϕ + = ) (

ˆ ˆ =

r

n Dµ

ˆ =

νρ µg

D

r r r r

n n g n A A ˆ ˆ 1 ˆ ˆ

µ µ µ

∂ × − =

(iii) Abelian decomposition

• f the gauge potential

* Decomposition of Lorentz connection **

3(

(2)/ (1)) SU U π ∈

→

ˆa

i

n

is unobservable topological degree which becomes dynamical in effective Faddeev-Niemi-Skyrme model /1998 am

e

??

• III. Minimal model of Lorentz gauge gravity with torsion

The main idea: Existence of the topological phase in the absence of torsion The most general P, CP invariant Lagrangian quadratic in curvature:

In constant curvature space-time, we consider linear in Kmcd equations of motion:

[ ]

acdb abcd db bd bd cdab abcd abcd

R R R R R R R R L γ β γ α γ α 4 ) ( 4 ) ( ) ( 4 1

2 2

+ − − − − + − =

d e b e c a d c b a mcd abcd mcd abcd abcd abcd

K K K D K e R K e R R R

] | [ ] [

ˆ ) ; ( ~ ) ; ( ~ ˆ + = + =

) ( 12 ˆ ˆ

bc ad bd ac abcd

R R η η η η − =

) 2 (

=

bcd

K L δ δ

Decomposition into irreducible field components

• Pure gauge degrees of freedom (6 dof):

K0cd Space components of contortion: two spin 2 states (4 dof): four vector spin 1 states (12 dof): two spin 0 states (2 dof): total number of physical dof: 18

σ µρσ µ ρ ρ µ ρ µ µρ µρ ρ µ σ µρσ µ ρ ρ µ ρ µ µρ µρ µρ µρ γδρ µγδ

ε δ ε δ ε Q R R R R K A S S S S K K K

t tt t tt

+ ∂ + ∂ + ∂ ∂ − Δ + = + ∂ + ∂ + ∂ ∂ − Δ + = = ) ( ) ( 2 1 , ) ( ) ( 2 1 ~ , ~

t t tt tt

R S Q R A S R S , , , , ,

µ µ µ µ µρ µρ

There are constraints in the eqns. of motion

Additional local symmetries of the equations of motion: We impose 6 gauge fixing conditions to fix the local Lorentz symmetry: (the gauge is consistent with equations of motion)

dc d db c bcd c bd d bc bcd

D D K D D K χ χ δ λ η λ η δ

χ λ

ˆ ˆ , ˆ ˆ 3 1 − = − =

) 2 (

=

bcd

K L δ δ

ˆ ,

,

= = =

cb c cc cb bc

D χ χ χ χ

, ) ( ) ( , ) ( = ∂ ∂ = ∂ − ∂ − ∂ + = − ∂

j i j i i i i i i i i i i

K K K K K K

γ δ δ γ γδ δ δ

γ γ α

Technical details

# we split the Lorentz connection into background and quantum parts. We consider linearized in Kmcd Euler- Lagrange equations of motion in the constant curvature background Riemannian space. # Normal gauge decomposition of geometric quantities is used in sufficient order of R. # the theory is highly degenerated, there appear constraints which strongly suppress the dynamics of torsion. We solve all constraints while keeping the consistence with dynamical eqns. of motion .

cd cd cd

K e A

µ µ µ

ϕ + = ) (

ˆ ( ) ... 36

a a a k a m m m k m

R e x x x x δ δ = + − +

The final result of calculation is the following:

Special case α=1, β=0, γ= -3: torsion obtains dynamical degrees All irreducible fields in the decomposition of contortion Kmcd are expressed in terms of four fields: massless vector, 2 d.o.f. The number of torsion dynamic spin 2, 2 d.o.f. d.o.f. equals the number of d.o.f. for the metric tensor! two spin 0 fields, 2 d.o.f. Classical effective Lagrangian is:

long long tt tr

Q S S A

µ µ µ µ µν µ

ϕ ϕ ∂ = ∂ =

2 1

( ) ( ) ( ) ( )

( )

2 2 1 2 2 2 2 1 2 2 2 1 2 2 2 2 2

6 6 1 1 8 3 2 ϕ ϕ ϕ ρ ϕ ϕ ρ ϕ ρ ρ

αβ αβ α α t t t tt t tt tr t tr eff

S S A A L ∂ + − − ∂ + ∂ − − − ∂ + ∂ − ∂ − ∂ + ∂ − ∂ + − ∂ + ∂ − = r r r r r r

ˆ 24 R ρ =

Covariant quantization in one-loop approximation

• Effective Lagrangian with

gauge fixing terms and Faddeev-Popov ghosts has a simple form:

cd cd FP FP cd cd FP

R D D L c D D c L c R D D c L ψ ψ ) 3 ˆ ˆ ˆ ( ˆ ˆ ) 6 ˆ ˆ ˆ (

) 3 ( 2 2 ) 2 ( 1 1 ) 1 (

− = = + =

∑

=

+ + =

3 , 2 , 1 ) ( ) ( ) 2 (

) ( ) , (

i i FP i gf class tot

L L K e L L

2 3 ) 3 ( 2 2 ) 2 ( 2 1 ) 1 (

) ~ ˆ ~ ˆ ( 2 1 ) ~ ˆ ( 2 1 ) ~ ˆ ( 2 1

dcb c bcd c gf d d gf bcd b gf

K D K D L K D L K D L + − = − = − = ξ ξ ξ

Conclusions

Drawbacks of Yang-Mills type Lorentz gauge gravity: * The Hamiltonian is not positively defined; * Two independent variables, vielbein and contortion, on equal footing. Vielbein should be quantized also. Which variable is fundamental? * The content of spin states of contortion is too big to compare with vielbein, 24 degrees of freedom; How are they resolved in the minimal gravity with torsion: * The kinetic terms for spin 1,2 are

• positive. There is a hope that the

Hamiltonian of whole non-linear theory is positive. * Torsion can be treated as a unique dynamic degree of quantum gravity. * The number of physical d.o.f. for torsion and for the metric tensor are the same. # Metric becomes dynamical in the effective Einstein gravity. Topological d.o.f. turn into dynamical ones.

Open questions

* Implications in standard cosmology: Dark matter as a classical or quantum condensate of torsion. * Idea from analogous gravity models in condensed matter: the torsion condensate provides microscopic structure of space as a superfluid. * If torsion does not exist as a classical object then the space before Big Bang is topological, non-metric one. * If there is no torsion at all then we’ll have a chance to invent more sophisticated and beautiful theory.